There are a number of ways to troubleshoot network connections on (or using) an iOS device. These can be common troubleshooting steps that you might run from the command line or a third party app on a desktop computer or they could be specific to testing the network environment for an iOS device. Some of these apps are even free.

Ping Lite
One of the most common tasks that most administrators routinely do to test both DNS resolution and connectivity is pinging something. Ping Lite comes with a function to show your IP, a ping tool, a tool to ping the subnet, the ability to run trace routes and for good measure a little telnet love as well. Not bad for the fat price of nothing. Developed by MochaSoft, Ping Lite is a must for anyone who does any kind of network troubleshooting, unless you’re paying good money for a more robust tool!

NSLookup
Ping Lite is a great tool for isolating whether you’re having connectivity problems to an IP address. However, if Exchange’s auto discover isn’t working or some other

Bonjour Browser
One of my favorite tools for finding things on the network, Bonjour is a multicast tool and what many of the features meant to be used in a home where zero configuration networking is important

Speed Test
I think that one of the more common tasks in troubleshooting network connections is to determine whether Internet speed is satisfactory. Satisfactory is a relative term. Both relative to the expected performance and relative to the perception of users. For example, the bandwidth that a user is getting on a device may exceed the expected performance based on the speed provided by the DSL, cable modem or other WAN connection provided. However, that speed may be less than what the user’s would like (one can never have enough bandwidth!).

ezShare
ezShare is a nice little tool that lets administrators log into shares of various types. The cool thing about this little tool is that you can connect via SSH, FTP, WebDAV, S3, Google Docs, Box.net, SMB/CIFS, or NFS. This allows you to test WebDAV from a different tool if you’re having a problem opening WebDAV connections from within Pages, test the speed of downloading a document from a FTP site, check Google Docs or Box.net connectivity and even see if that file server is available when users call in with problems connecting to SMB/CIFS shares on Windows servers.

Bonus App:
AirPort Utility
If you have an Apple AirPort acting as a WAP or the gateway to your office/home then this little app is awesome. Apple has eased the setup process for their Wireless Access Points to the point that you can set the entire thing up, change settings and even troubleshoot the odd connectivity issue without ever touching a desktop computer. AirPort Utility is also a great way to test whether you can connect to shares hosted by devices and update passwords on the fly.